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Eigenvalue estimates and nodal length of eigenfunctions. (English) Zbl 1055.58015

Kozma, L. (ed.) et al., Steps in differential geometry. Proceedings of the colloquium on differential geometry, Debrecen, Hungary, July 25–30, 2000. Debrecen: Univ. Debrecen, Institute of Mathematics and Informatics. 295-301 (2001).
The paper under review is a nice and clear survey (without proof) of the following result: Let \(M\) be a 2-dimensional compact, smooth Riemannian manifold without boundary, and let \( \Phi \) be an eigenfunction associated to the eigenvalue \( \lambda \). Then the bound: \[ \text{Length}[\Phi ^{-1}(0)] > \frac{1}{11} \text{Area}(M) \sqrt{\lambda} \] holds if \( \lambda \) is large. If the curvature of \( M \) is everywhere non-negative, then the bound holds for all eigenvalues.
Details can be found in Ann. Global Anal. Geom. 19, No. 2, 133–151 (2001; Zbl 1010.58025).
For the entire collection see [Zbl 0966.00031].

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
53C20 Global Riemannian geometry, including pinching

Citations:

Zbl 1010.58025
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